\(\int (a+b \sec (e+f x))^m \, dx\) [791]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int (a+b \sec (e+f x))^m \, dx=\text {Int}\left ((a+b \sec (e+f x))^m,x\right ) \]

[Out]

Unintegrable((a+b*sec(f*x+e))^m,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \sec (e+f x))^m \, dx=\int (a+b \sec (e+f x))^m \, dx \]

[In]

Int[(a + b*Sec[e + f*x])^m,x]

[Out]

Defer[Int][(a + b*Sec[e + f*x])^m, x]

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (e+f x))^m \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.65 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (e+f x))^m \, dx=\int (a+b \sec (e+f x))^m \, dx \]

[In]

Integrate[(a + b*Sec[e + f*x])^m,x]

[Out]

Integrate[(a + b*Sec[e + f*x])^m, x]

Maple [N/A] (verified)

Not integrable

Time = 0.60 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \left (a +b \sec \left (f x +e \right )\right )^{m}d x\]

[In]

int((a+b*sec(f*x+e))^m,x)

[Out]

int((a+b*sec(f*x+e))^m,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e) + a)^m, x)

Sympy [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (e+f x))^m \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

integrate((a+b*sec(f*x+e))**m,x)

[Out]

Integral((a + b*sec(e + f*x))**m, x)

Maxima [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^m, x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int (a+b \sec (e+f x))^m \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sec(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^m, x)

Mupad [N/A]

Not integrable

Time = 14.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.33 \[ \int (a+b \sec (e+f x))^m \, dx=\int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int((a + b/cos(e + f*x))^m,x)

[Out]

int((a + b/cos(e + f*x))^m, x)